| L(s) = 1 | − 0.485·2-s − 1.76·4-s + 5-s − 1.61·7-s + 1.82·8-s − 0.485·10-s + 4.95·13-s + 0.785·14-s + 2.64·16-s + 3.13·17-s + 1.09·19-s − 1.76·20-s + 6.75·23-s + 25-s − 2.40·26-s + 2.85·28-s − 3.55·29-s + 8.53·31-s − 4.93·32-s − 1.52·34-s − 1.61·35-s − 5.35·37-s − 0.532·38-s + 1.82·40-s + 5.32·41-s + 3.41·43-s − 3.27·46-s + ⋯ |
| L(s) = 1 | − 0.343·2-s − 0.882·4-s + 0.447·5-s − 0.612·7-s + 0.645·8-s − 0.153·10-s + 1.37·13-s + 0.210·14-s + 0.660·16-s + 0.761·17-s + 0.251·19-s − 0.394·20-s + 1.40·23-s + 0.200·25-s − 0.471·26-s + 0.540·28-s − 0.659·29-s + 1.53·31-s − 0.872·32-s − 0.261·34-s − 0.273·35-s − 0.881·37-s − 0.0864·38-s + 0.288·40-s + 0.830·41-s + 0.520·43-s − 0.483·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.503943214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503943214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.485T + 2T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 - 3.13T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 - 8.53T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 - 5.32T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 + 9.71T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 + 6.63T + 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 - 8.49T + 73T^{2} \) |
| 79 | \( 1 + 1.99T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.273808789726818483643018653279, −7.63226254364898488535360556842, −6.65351690800103865702300472981, −6.05015852634688642136107508976, −5.26756011040392220272477086529, −4.56695693955541506978786740506, −3.55347901304610147340423644549, −3.05055444357573052333481860354, −1.56199454104429717548767759430, −0.75976703878265416758858988899,
0.75976703878265416758858988899, 1.56199454104429717548767759430, 3.05055444357573052333481860354, 3.55347901304610147340423644549, 4.56695693955541506978786740506, 5.26756011040392220272477086529, 6.05015852634688642136107508976, 6.65351690800103865702300472981, 7.63226254364898488535360556842, 8.273808789726818483643018653279
